On Poincare-Friedrichs Type Inequalities for the Broken Sobolev Space W2,1

We are concerned with the derivation of Poincare-Friedrichs type inequalities in the broken Sobolev space W-2,W-1(Omega; T-h) with respect to a geometrically conforming, simplicial triagulation T-h of a bounded Lipschitz domain Omega in R-d, d is an element of N. Such inequalities are of interest in the numerical analysis of nonconforming finite element discretizations such as C-0 Discontinuous Galerkin (C(0)DG) approximations of minimization problems in the Sobolev space W-2,W-1(Omega), or more generally, in the Banach space BV2(Omega) of functions of bounded second order total variation. As an application, we consider a C(0)DG approximation of a minimization problem in BV2(Omega) which is useful for texture analysis and management in image restoration.